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Edexcel Maths FP3


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Topic Questions from Papers


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Vectors

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4

*M35145A0428*

2.

O

A

B

Ca c

b

Figure 1

The points A, B and C have position vectors a, b and c respectively, relative to a fixed origin O, as shown in Figure 1.

It is given thata = i + j, b = 3i – j + k and c = 2i + j – k.

Calculate

(a) b c,(3)

(b) a . (b c),(2)

(c) the area of triangle OBC,(2)

(d) the volume of the tetrahedron OABC.(1)

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×

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Question 2 continued

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(Total 8 marks)

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*M35145A02028*

7. The lines 1l and 2l have equations

r = + and r = + .

If the lines 1l and 2l intersect, find

(a) the value of ,(4)

(b) an equation for the plane containing the lines 1l and 2l , giving your answer in the form ax + by + cz + d = 0, where a, b, c and d are constants.

(4)

For other values of , the lines 1l and 2l do not intersect and are skew lines.

Given that = 2,

(c) find the shortest distance between the lines 1l and 2l .(3)

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−⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

134

112

−⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

α−⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

40

032

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

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*N35389RA02028*

7. The plane has vector equation

r = 3i + k + (–4i + j) + (6i – 2j + k)

(a) Find an equation of in the form r.n p= , where n is a vector perpendicular to and is a constant.

(5)

The point has coordinates (6, 13, 5). The line passes through and is perpendicular to . The line intersects at the point .

(b) Show that the coordinates of are (3, 1, –1).(4)

The point lies on and has coordinates (1, 0, 2).

(c) Find the perpendicular distance from to the line . Give your answer to 3 significant figures.

(5)

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*P35414A01628*

6. The plane P has equation

(a) Find a vector perpendicular to the plane P.(2)

The line l passes through the point A (1, 3, 3) and meets P at (3, 1, 2).

The acute angle between the plane P and the line l is .

(b) Find to the nearest degree.(4)

(c) Find the perpendicular distance from A to the plane P.(4)

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r =⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

+−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

+⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

312

021

322

λ μ

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*P40111A0632*

3. The position vectors of the points A, B and C relative to an origin O are i j k− −2 2 , 7 3i k− and 4 4i j+ respectively.

Find

(a) AC BC× ,(4)

(b) the area of triangle ABC,(2)

(c) an equation of the plane ABC in the form r. n = p(2)

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*P42956A0432*

2. Two skew lines l1 and l2 have equations

l1: r = (i – j + k) + �(4i + 3j + 2k)

l2: r = (3i + 7j + 2k) + �(–4i + 6j + k)

respectively, where � and � are real parameters.

(a) Find a vector in the direction of the common perpendicular to l1 and l2(2)

(b) Find the shortest distance between these two lines.(5)

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*P43143A02428*

8. The plane �1 has vector equation

r.(3i – 4j + 2k) = 5

(a) Find the perpendicular distance from the point (6, 2, 12) to the plane �1

(3)

The plane �2 has vector equation

r = �(2i + j + 5k) + �(i – j – 2k), where � and � are scalar parameters.

(b) Find the acute angle between �1 and �2 giving your answer to the nearest degree.(5)

(c) Find an equation of the line of intersection of the two planes in the form r × a = b, where a and b are constant vectors.

(6)

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 14 marks)

10 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2009

Further Pure Mathematics FP3 Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1–C4.

Vectors

The resolved part of a in the direction of b is b

a.b

The point dividing AB in the ratio μλ : is μλλμ

++ ba

Vector product: −−−

===×

1221

3113

2332

321

321ˆ sinbabababababa

bbbaaakji

nbaba θ

)()()(

321

321

321

bac.acb.cba. ×=×==×cccbbbaaa

If A is the point with position vector kjia 321 aaa ++= and the direction vector b is given by

kjib 321 bbb ++= , then the straight line through A with direction vector b has cartesian equation

)( 3

3

2

2

1

1 λ=−

=−

=−

baz

bay

bax

The plane through A with normal vector kjin 321 nnn ++= has cartesian equation

a.n−==+++ ddznynxn where0321

The plane through non-collinear points A, B and C has vector equation

cbaacabar μλμλμλ ++−−=−+−+= )1()()( The plane through the point with position vector a and parallel to b and c has equation

cbar ts ++=

The perpendicular distance of ) , ,( γβα from 0321 =+++ dznynxn is 23

22

21

321

nnn

dnnn

++

+++ γβα.

Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 11

Hyperbolic functions 1sinhcosh 22 =− xx

xxx coshsinh22sinh = xxx 22 sinhcosh2cosh +=

)1( 1lnarcosh }{ 2 ≥−+= xxxx

}{ 1lnarsinh 2 ++= xxx

)1( 11lnartanh 2

1 <−+= x

xxx

Conics

Ellipse Parabola Hyperbola Rectangular

Hyperbola

Standard Form 12

2

2

2

=+by

ax axy 42 = 12

2

2

2

=−by

ax 2cxy =

Parametric Form

)sin ,cos( θθ ba )2 ,( 2 atat (a sec θ, b tan θ ) (±a cosh θ, b sinh θ ) t

cct,

Eccentricity 1<e

)1( 222 eab −= 1=e 1>e

)1( 222 −= eab e = √2

Foci )0 ,( ae± )0 ,(a )0 ,( ae± (±√2c, ±√2c)

Directrices eax ±= ax −=

eax ±= x + y = ±√2c

Asymptotes none none by

ax ±= 0 ,0 == yx

12 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009

Differentiation

f(x) f ′(x)

xarcsin 21

1

x−

xarccos 21

1

x−−

xarctan 211x+

xsinh xcosh xcosh xsinh xtanh x2sech

xarsinh 21

1

x+

xarcosh 1

12 −x

artanh x 211x−

Integration (+ constant; 0>a where relevant)

f(x) xx d)f(

xsinh xcosh xcosh xsinh xtanh xcoshln

22

1

xa − )( arcsin ax

ax <

221

xa +

ax

aarctan1

22

1

ax − )( ln,arcosh }{ 22 axaxx

ax >−+

22

1

xa + }{ 22ln,arsinh axx

ax ++

221

xa − )( artanh1ln

21 ax

ax

axaxa

a<=

−+

221

ax −

axax

a +−ln

21

Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1–September 2009 13

Arc length

xxys d

dd1

2

+= (cartesian coordinates)

tty

txs d

dd

dd 22

+= (parametric form)

Surface area of revolution

Sx = sy d2π = + xxyy d

dd12

2

π

= + tty

txy d

dd

dd2

22

π

8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009

Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2.

Summations

)12)(1(61

1

2 ++==

nnnrn

r

2241

1

3 )1( +==

nnrn

r

Numerical solution of equations

The Newton-Raphson iteration for solving 0)f( =x : )(f)f(

1n

nnn x

xxx′

−=+

Conics

Parabola Rectangular

Hyperbola

Standard Form axy 42 = xy = c2

Parametric Form (at2, 2at) t

cct,

Foci )0 ,(a Not required

Directrices ax −= Not required

Matrix transformations

Anticlockwise rotation through θ about O: −

θθθθ

cos sinsincos

Reflection in the line xy )(tanθ= : − θθ

θθ2cos2sin2sin 2cos

In FP1, θ will be a multiple of 45°.

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 – Issue 1 – September 2009 7

Core Mathematics C4 Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3.

Integration (+ constant)

f(x) xx d)f(

sec2 kx k1 tan kx

xtan xsecln

xcot xsinln

xcosec )tan(ln,cotcosecln 21 xxx +−

xsec )tan(ln,tansecln 41

21 π++ xxx

−= xxuvuvx

xvu d

ddd

dd

6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2009

Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2.

Logarithms and exponentials xax a=lne

Trigonometric identities

BABABA sincoscossin)(sin ±=± BABABA sinsincoscos)(cos =±

))(( tantan1tantan)(tan 2

1 π+≠±±=± kBABABABA

2cos

2sin2sinsin BABABA −+=+

2sin

2cos2sinsin BABABA −+=−

2cos

2cos2coscos BABABA −+=+

2sin

2sin2coscos BABABA −+−=−

Differentiation

f(x) f ′(x) tan kx k sec2 kx sec x sec x tan x cot x –cosec2 x cosec x –cosec x cot x

)g()f(

xx

))(g()(g)f( )g()(f

2xxxxx ′−′

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5

Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rule

a2 = b2 + c2 – 2bc cos A

Binomial series

2

1

)( 221 nrrnnnnn bbarn

ban

ban

aba ++++++=+ −−− (n ∈ )

where )!(!

!C rnr

nrn

rn

−==

∈<+×××

+−−++×−++=+ nxx

rrnnnxnnnxx rn ,1(

21)1()1(

21)1(1)1( 2 )

Logarithms and exponentials

axx

b

ba log

loglog =

Geometric series un = arn − 1

Sn = r ra n

−−

1)1(

S∞ = r

a−1

for ⏐r⏐ < 1

Numerical integration

The trapezium rule: b

axy d ≈ 2

1 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where n

abh −=

4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009

Core Mathematics C1

Mensuration

Surface area of sphere = 4π r 2

Area of curved surface of cone = π r × slant height

Arithmetic series un = a + (n – 1)d

Sn = 21 n(a + l) =

21 n[2a + (n − 1)d]

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Combined QP - FP3 Edexcel - PMT